Let $E/F$ be a quadratic field extension with Galois group $\{1,c\}$, where $c$ is the nontrivial automorphism of $E$. Consider $V$ to be a $E$-vector space of dimension $n$. We equip it with a $\epsilon$-Hermitian form $\phi: V \times V \rightarrow E$. Then in Page 90 of the book of Mœglin-Vignéras-Waldspurger, it is claimed that there exists an element $\delta \in \mathrm{GL}(V)$ (not necessarily preserving the form $\phi$) such that $\phi(\delta v, \delta w) = \phi(w,v)$ for any $v,w \in V$.
My question is:
- How to prove the existence of such a $\delta$? It is claimed in Page 90 of the book of MVW's book that this is easy, but I cannot see how. I have been trying to play with matrices, but failed. It is also claimed in Page 79 of MVW's book that this is proved in Section IV.2 of this article, but I cannot find the precise place of the demanding statement and its proof.
- Can I write the element $\delta$ explicity as a matrix when an $E$-basis of $V$ is fixed. In particular, I am interested in a particular basis of $E$ of the form $\{e_1, \ldots, e_s, w_1, \ldots, w_r, f_1, \ldots, f_r\}$ such that $\phi$ is of the form $$ \phi = \begin{pmatrix} 0 & 0 & 1_{s} \\ 0 & \vartheta & 0 \\ -1_{s} & 0 & 0 \end{pmatrix}, $$ where $\vartheta$ is a square matrix of size $r$ such that $({\vartheta^{c}})^{\mathrm{t}} = - \vartheta$. So the form $\phi$ is $(-1)$-Hermitian, i.e. $(\phi^{c})^{\mathrm{t}} = -\phi$. I have also been trying to construct seemingly obvious matrices, but none of my attempts give the demanding $\delta$.
If such an element $\delta$ is found, then we have an outer automorphism of the unitary group $G := \mathrm{U}(V)$, given by $g \mapsto \delta g \delta^{-1}$. This is useful in the theory of smooth representations of $G(F_v)$ where $v$ is a nonarchimedean place of $F$, $E/F$ is a CM extension. This is my motivation for the question.